research-article

Self-Similar Fractal Drawings Inspired by M. C. Escher’s Print Square Limit

Authors:

Peichang Ouyang,

Kwok Wai Chung,

Krzysztof GdawiecAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 40, Issue 3

Article No.: 31, Pages 1 - 34

https://doi.org/10.1145/3456298

Published: 15 July 2021 Publication History

Abstract

A fractal tiling (f-tiling) is a kind of rarely explored tiling by similar polygonal tiles which possesses self-similarity and the boundary of which is a fractal. Based on a tiling by similar isosceles right triangles, Dutch graphic artist M. C. Escher created an ingenious print Square Limit in which fish are uniformly reduced in size as they approach the boundaries of the tiling. In this article, we present four families of f-tilings and propose an easy-to-implement method to achieve similar Escher-like drawings. By systematically investigating the local star-shaped structure of f-tilings, we first enumerate four families of f-tilings admitted by kite-shaped or dart-shaped prototiles. Then, we establish a fast binning algorithm for visualising f-tilings. To facilitate the creation of Escher-like drawings on the reported f-tilings, we next introduce one-to-one mappings between the square, and kite and dart, respectively. This treatment allows a pre-designed square template to be deformed into all prototiles considered in the article. Finally, we specify some technical implementations and present a gallery of the resulting Escher-like drawings. The method established in this article is thus able to generate a great variety of exotic Escher-like drawings.

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Cited By

Alam KRohen YSaleem NAphane MRazzaque A(2025)On escape criterion of an orbit with s−convexity and illustrations of the behavior shifts in Mandelbrot and Julia set fractalsPLOS ONE10.1371/journal.pone.031219720:1(e0312197)Online publication date: 7-Jan-2025
https://doi.org/10.1371/journal.pone.0312197
Ouyang PChung KFathauer RNicolas APang JCao SGdawiec K(2025)Generation of Escher-Like Rosette DrawingsJournal of Computer Science and Technology10.1007/s11390-024-2874-539:6(1466-1479)Online publication date: 16-Jan-2025
https://doi.org/10.1007/s11390-024-2874-5
Almutlg AAhmad I(2024)Fractals as Julia Sets for a New Complex Function via a Viscosity Approximation Type Iterative MethodsAxioms10.3390/axioms1312085013:12(850)Online publication date: 3-Dec-2024
https://doi.org/10.3390/axioms13120850
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Index Terms

Self-Similar Fractal Drawings Inspired by M. C. Escher’s Print Square Limit
1. Applied computing
  1. Arts and humanities
    1. Fine arts
2. Computing methodologies
  1. Computer graphics

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Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 40, Issue 3

June 2021

264 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/3463476

Editor:
Marc Alexa
TU Berlin, Germany

Issue’s Table of Contents

Copyright © 2021 Association for Computing Machinery.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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Publication History

Published: 15 July 2021

Accepted: 01 March 2021

Revised: 01 February 2021

Received: 01 December 2019

Published in TOG Volume 40, Issue 3

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Natural Science Foundation of China
Science and Technology Project of the Education Department of Jiangxi Province of China
Natural Science Foundation of Jiangxi Province of China

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Cited By

Alam KRohen YSaleem NAphane MRazzaque A(2025)On escape criterion of an orbit with s−convexity and illustrations of the behavior shifts in Mandelbrot and Julia set fractalsPLOS ONE10.1371/journal.pone.031219720:1(e0312197)Online publication date: 7-Jan-2025
https://doi.org/10.1371/journal.pone.0312197
Ouyang PChung KFathauer RNicolas APang JCao SGdawiec K(2025)Generation of Escher-Like Rosette DrawingsJournal of Computer Science and Technology10.1007/s11390-024-2874-539:6(1466-1479)Online publication date: 16-Jan-2025
https://doi.org/10.1007/s11390-024-2874-5
Almutlg AAhmad I(2024)Fractals as Julia Sets for a New Complex Function via a Viscosity Approximation Type Iterative MethodsAxioms10.3390/axioms1312085013:12(850)Online publication date: 3-Dec-2024
https://doi.org/10.3390/axioms13120850
Schor AKim T(2024)Into the Portal: Directable Fractal Self-SimilarityACM SIGGRAPH 2024 Conference Papers10.1145/3641519.3657466(1-8)Online publication date: 13-Jul-2024
https://dl.acm.org/doi/10.1145/3641519.3657466
Nagata YImahori S(2024)Creation of Dihedral Escher-like Tilings Based on As-Rigid-As-Possible DeformationACM Transactions on Graphics10.1145/363804843:2(1-18)Online publication date: 22-Jan-2024
https://dl.acm.org/doi/10.1145/3638048
Yin ZJin YFang ZZhang YZhang HZhou JHe L(2024)Symmetrization of quasi-regular patterns with periodic tilting of regular polygonsComputational Visual Media10.1007/s41095-023-0359-z10:3(559-576)Online publication date: 27-Apr-2024
https://doi.org/10.1007/s41095-023-0359-z
Anagha PGopan G. S. SAbraham N(2023)Simulation Studies on the Design and Analysis of Interdigital and Fractal-Based Micro-SupercapacitorsECS Journal of Solid State Science and Technology10.1149/2162-8777/acdd9912:6(061003)Online publication date: 20-Jun-2023
https://doi.org/10.1149/2162-8777/acdd99
S. SAbraham NS REdison A(2023)Investigations on Recent Advancements in the Fabrication of In-Plane Micro-Supercapacitors as Portable Energy Storage Devices2023 International Conference on Control, Communication and Computing (ICCC)10.1109/ICCC57789.2023.10165267(1-6)Online publication date: 19-May-2023
https://doi.org/10.1109/ICCC57789.2023.10165267
Kumari SGdawiec KNandal AKumar NChugh R(2023)On the viscosity approximation type iterative method and its non-linear behaviour in the generation of Mandelbrot and Julia setsNumerical Algorithms10.1007/s11075-023-01644-496:1(211-236)Online publication date: 17-Aug-2023
https://dl.acm.org/doi/10.1007/s11075-023-01644-4
Chung KOuyang PNicolas ACao SBailey DGdawiec K(2023)Generation of Escher‐like spiral drawings in a modified hyperbolic spaceMathematical Methods in the Applied Sciences10.1002/mma.933246:13(14489-14508)Online publication date: 6-May-2023
https://doi.org/10.1002/mma.9332
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